The invention relates to a method for tilting both an optical window having a small wedge angle and a planar return flat so as to cancel "ripple" or "ghost fringes" in the intensity pattern of the wavefront which is transmitted by the window and then returned by the planar return flat. The ghost fringes occur as a result of both multiple internal reflections within the window and the small wedge angle. The optical window can be a plane parallel plate or a dome-shaped window with opposed surfaces having the same center of curvature.
Spurious reflections usually introduce errors into measurement results obtained with laser phase-shifting interferometry. Work has been done for a Fizeau interferometer to reduce or eliminate the effect of the multiple reflections between a test surface and a reference surface. It is known that if a four-frame phase calculation algorithm is used, the phase error caused by multiple reflections is eliminated to a first order approximation. A new algorithm is known which can completely eliminate the phase error due to multiple reflections of a test mirror. However, no one has been able to eliminate "ripple" errors due to very small wedge angles of an optical window, i.e., of a plane parallel plate.
Multiple reflections between two surfaces of a window introduce a fixed pattern error in the transmitted wavefront. In a Fizeau or Twyman-Green interferometer this wavefront is reflected by a return flat and transmitted back through the window. The fixed pattern error or ghost fringe pattern is carried in the measurement result. The ghost fringe pattern error is negligible only if the wedge angle is so large that the interference fringes in the interference pattern representing the transmission characteristics of the window are too dense for the detector to resolve. However, if the wedge angle is small (e.g. several arc-seconds), the phase error could be up to 0.025 fringes for most glass. The measurement of a precision optical window requires an accuracy of the transmitted wavefront within 0.1 fringe or smaller. Therefore, the 0.025 fringe error caused by the multiple reflections significantly effects the accuracy of the measurement. The size of wedge angle that causes appreciable ripple depends on the diameter of the window. For example, for a 1 inch window a wedge angle of 2 arc-seconds will produce one ghost fringe, but for a 10 inch window a wedge angle of only 0.2 arc-seconds will produce one ghost fringe.
FIG. 1 shows a planar parallel plate or optical window 1 being testing in a conventional interferometer 33, such as a WYKO 6000. The relative amplitudes of the successive internally reflected rays such as 2C and 2F are 1, r.sup.2, r.sup.4, . . . , where r is the coefficient of reflection of window 1. If the incident angle (i.e., the angle between line 3, which is perpendicular to window surface 1A, and incident beam 2A) is .theta., it can be shown that the optical path difference (OPD) of two successive rays is equal to 2dn cos(.theta.), where d and n are the thickness and the refractive index of window 1, respectively, and .theta. is the refractive angle between the perpendicular line 3 and refracted ray 2B in FIG. 1A. It should be appreciated that window 1 must be tilted at an incident angle of .theta. to prevent a portion of rays multiply reflected by surfaces 1A and 1B and spuriously transmitted rays from returning into the detector 30 of interferometer 33 of FIG. 1. The coefficient of reflection (r) of most optical glass is about 20%. Therefore the multiple reflections of a window can be approximated by the first two rays, i.e., 2C and 2F, with reflective amplitudes of 1 and r.sup.2, respectively.
"Composite" wavefront 5A,B and the measurement results have "ripples" in intensity 6 of the interferogram perpendicular to the wedge direction x. The transmitted wavefront 5A has intensity levels much greater than wavefront 5B. Nevertheless, the composite wavefront 5A,B has an appreciable ripple or ghost fringe pattern caused by interference of wavefront 5B with wavefront 5A. If the primary incident beam 2A and the secondary beam 2F are not parallel due to a minute wedge angle between window surfaces 1A and 1B, then the transmitted composite wavefront 5A,B will have a ripple or ghost fringe pattern in its interferogram. The density of ghost fringes is proportional to the wedge angle of the window. For large wedge angles, the density of fringes is so great that it is undetectable and does not cause appreciable error in measurements.
As shown in FIG. 2, the two rays 2C and 2F are reflected back by a return flat 9 as rays 11A and 10A, respectively, to window 1. The coefficient of reflection of return flat 9 is s, as shown in FIG. 1. Between window 1 and return flat 9 the filled-arrow ray 11A has a relative amplitude of s, and the open-arrow ray 10A has a relative amplitude of r.sup.2 s. Each of the two rays 11A and 10A reflected by return flat 9 undergoes multiple reflections within window 1. Due to the multiple reflections 11D and 11E of the primary "filled-arrow" ray 11B in window 1, the transmitted ray can be approximated by the first two rays 11C and 11F, E.sub.t and E.sub.g2. Because of the low reflectivity of window 1, the multiple reflections of the "open-arrow" ray 10B inside window 1 are negligible, and only the transmitted ray E.sub.g1 is significant. For an incident ray 2A entering window 1, there are three returned rays, E.sub.t, E.sub.g1, and E.sub.g2, as shown in FIG. 2. Because of the non-zero incident angle, the returned rays E.sub.g1 and E.sub.g2 are laterally displaced from the original incident location by approximately d.theta.(1-1/n), and go through different regions x.sub.1 and x.sub.2, respectively, of the window. If the thicknesses of the two regions are d(x.sub.1,y) and d(x.sub.2,y), respectively, the complex amplitudes of the three returned rays are EQU E.sub.t =s expi[2.phi..sub.w (x,y)+.phi..sub.r (x,y)], (1.1) EQU E.sub.g1 =r.sup.2 s expi[.phi..sub.w (x,y)+.phi..sub.w (x.sub.1, y)+.phi..sub.r (x,y)+2d(x.sub.1,y)n cos(.theta.')k], (1.2) EQU E.sub.g2 =r.sup.2 s expi[.phi..sub.w (x,y)+.phi..sub.w (x.sub.2,y)+.phi..sub.r (w,y)+2d(x.sub.2,y)n cos(.theta.')k],(1.3)
where k=2.pi./.lambda., and .theta.' is the refracted angle inside the window. .phi..sub.w (x,y) and .phi..sub.r (x,y) are the contributions of window 1 and return flat 9, respectively, to the complex amplitudes of the returned rays.
For a thin window or small incident angle, the displacement is negligible, i.e., x.sub.1 .apprxeq.x.sub.2.
In FIG. 2, return flat 9 is not tilted (i.e., is perpendicular to the incident ray or optical axis of the interferometer), so x.sub.1 .apprxeq.x.sub.2 .apprxeq.x. The "wedge angle" is the angle between the opposed surfaces 1A and 1B of the window 1 in the x direction, i.e., the wedge direction. Because of the wedge angle, the optical thickness d(x,y)n is not constant over the area of window 1. The phase of the vector sum of the phasors of the three returned rays E.sub.t, E.sub.g1, and E.sub.g2 is a function of d(x,y)n, with a period of .lambda.. Because the vector sum of the three phasors E.sub.t, E.sub.g1, and E.sub.g2 varies with the optical thickness of window 1 along its wedge direction, i.e., the x direction, the resulting wavefront has ripples. An important example of the problems caused by such ripples is that they have prevented accurate measurement of windows used in large lasers, wherein windows with extremely flat, parallel surfaces are required.
There has long been an unmet need for a technique to eliminate inaccuracies caused by ghost fringes in the transmitted wavefronts of an optical window having a minute wedge angle. (One arc-second is one thirty-six hundredth of a degree.)